This tutorial is directed mainly towards beginning researchers, from
computer science, applied mathematics, and fields of science and
engineering in which partial differential equations play a major role. The
software centerpiece will be the Portable, Extensible Toolkit for
Scientific Computing (PETSc), freely available from Argonne National
Laboratory and part of the Terascale Optimal PDE Simulations (TOPS)
software project of the U.S. Department of Energy. Architecturally and
mathematically scalable linear iterative methods and nonlinearly implicit
methods for PDE systems will be presented and illustrated, and their use
in auxiliary problems, such as PDE-constrained optimization and
eigenanalysis will be pointed out.